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Present Value and Future Value Formulas

The small set of formulas behind every valuation, loan, and savings plan: single sums, regular payments, payments that last forever, rates that compound more than once a year, and the adjustment for inflation.

Lesson 7 of 8 · 7 min read · Advanced · Members · Last reviewed October 2026

The short version

Almost all financial arithmetic comes from one idea: money grows by a multiplier each period, so moving it forward in time means multiplying and moving it back means dividing. Five formulas cover most uses. One handles a single amount. Two handle a stream of equal payments, looking back to its value today or forward to what it builds up to. Two handle payments that go on forever, level or growing. Add the conversion between yearly and monthly rates and the adjustment for inflation, and you can price a bond, work out a loan payment, test a savings plan, and check any valuation in this course. The formulas are exact. The rates you put in them are estimates.

By the end of this lesson, you'll be able to

  • Move a single amount forward or back in time, and solve for the rate or the number of years
  • Value a stream of level payments, a perpetuity, and a growing perpetuity
  • Convert between rates for different periods and between returns before and after inflation

Why this matters

Time Value of Money and Discount Rates introduced the idea that a dollar today is worth more than a dollar later, and Discounted Cash Flow, Step by Step used it to value a company. Those lessons brought in formulas as they were needed. This one sets the whole toolkit out in one place, with a worked example of each, so you can recognize which formula a problem calls for.

One amount

Future value = Present value × (1 + rate)years

Present value = Future value ÷ (1 + rate)years

$10,000 invested at 6% for 20 years grows to $10,000 × 1.0620 = $32,071. Going the other way, $50,000 due in 15 years is worth $50,000 ÷ 1.0715 = $18,122 today at a 7% rate.

The same equation can be solved for either of the other two pieces.

  • The rate. If $10,000 became $25,000 in 12 years, the yearly rate was (25,000 ÷ 10,000)1 ÷ 12 − 1 = 7.9%. That's the geometric average from Arithmetic and Geometric Averages.
  • The time. At 6%, money doubles in about 11.9 years. The Rule of 72 from Compounding, 72 ÷ 6 = 12, is a quick approximation of the same thing.

Rates for periods shorter than a year

Interest is often added monthly or daily. A rate quoted as 6% a year, with interest added monthly, means 0.5% each month, and those months compound.

Effective yearly rate = (1 + yearly rate ÷ periods)periods − 1

(1.005)12 − 1 = 6.17%. The quoted 6% understates what's earned or owed. The gap grows with the rate: 1.5% a month is 18% a year by simple multiplication and 19.6% once the months compound.

The rule for every formula in this lesson: the rate and the number of periods have to be in the same unit. Monthly payments need a monthly rate and a count of months.

A stream of equal payments

A fixed payment each period for a set number of periods is called an . Loans, bond coupons, and regular savings all have this shape. You could discount each payment separately and add them up. The formulas do it in one step.

Present value = Payment × (1 − (1 + rate)−periods) ÷ rate

Future value = Payment × ((1 + rate)periods − 1) ÷ rate

Worked example: four questions with the same shape (hypothetical)
QuestionCalculationAnswer
What's $1,000 a year for 20 years worth today, at 6%?$1,000 × 11.470$11,470
What does saving $5,000 a year for 30 years build up to, at 6%?$5,000 × 79.058$395,291
What's the monthly payment on a $300,000 loan over 30 years at 6%?$300,000 ÷ 166.79, using 0.5% a month for 360 months$1,799
How much a year can $500,000 pay out for 30 years, earning 4%?$500,000 ÷ 17.292$28,915

The first two apply the formulas directly. The last two turn the present-value formula round: when you know the value today and want the payment, divide by the same factor. The loan's 360 payments add up to about $647,500, more than twice the amount borrowed. Payments are assumed to come at the end of each period, and the rates are assumptions for the arithmetic.

The rest of this lesson is for members

Still to come in this lesson: Payments that never end, Putting them together: a bond, Time makes rates matter more, Inflation, Which formula when, the key takeaways, and 2 quiz questions with explanations.

Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.

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